Why Is My Guitar String Frequency Calculation Different from the Model Answer?

[Philip Wong]

Homework Statement

string on a guitar is plucked to excites 3 harmonic.
length = 0.3m.
tension = 100N
linear mass density (mu) = 0.1 kg/m
calculate its frequency
My friend and I wants to find the frequency using the above information. I think we use different approach, but came to the same solution. YET when we compare it with model answers, it is WRONG (i.e. the model answer is 316Hz) can someone explain why? or was the model answer WRONG? (note: it is an known fact that the model answer is occasionally found to be wrong)

thanks!

Homework Equations

velocity = sqrt (T/mu)
L = 3/2 lambda i.e. lambda = (2/3) * L
velocity = lambda * frequency i.e. frequency = velocity/lambda

The Attempt at a Solution

velocity = sqrt(100/.1) =31.6 m/s
lambda = (2/3) * 0.3m = 0.2
frequency = 1/0.2 * 31.6 = 158Hz

the answer I got correlates with the answer a friend got too, but is was off by almost 200Hz when compare to model answer. SO it is definitely not due to rounding error.

Was we wrong? or was model answer wrong?

thanks!

 

[kuruman]

You and your friend are correct and the model answer is wrong. If you do it symbolically, the nth string harmonic frequency is given by

f_n=\frac{n}{2L}\sqrt{T/\mu}

Your answer is exactly half the model answer, so whoever wrote the model answer probably forgot to divide by 2.

 

[Philip Wong]

You and your friend are correct and the model answer is wrong. If you do it symbolically, the nth string harmonic frequency is given by

f_n=\frac{n}{2L}\sqrt{T/\mu}

Your answer is exactly half the model answer, so whoever wrote the model answer probably forgot to divide by 2.

thanks!